canonical setting

Weakly smooth log-concave

$V$ convex, $\lVert\nabla V(x)-\nabla V(y)\rVert\le\beta_s\lVert x-y\rVert^s$ , $s\in[0,1]$ .

Updated Aug 6, 2026

Upper and lower bounds

Comparison table

4 scoped results. Tildes suppress logarithmic factors.

Best-known and comparison results for Weakly smooth log-concave
Result	Algorithm or model	Complexity	Guarantee	Oracle / start	Assumptions and notes	Review	Sources
best upper	FORS + proximal sampler	$\widetilde O(\beta_s^{2/(1+s)}d^{s/(1+s)}R_0^2/\varepsilon^2)$	$\operatorname{KL}\le\varepsilon^2$	First-order + proximal implementation $W_2(\mu_0,\pi)=R_0$	Hölder gradient and the implementation assumptions in the theorem.Provisional transcription. At $s=0$ it gives the displayed nonsmooth endpoint.	Citedpreprint	Chen et al. (2026), Theorem G.1(3) ↗
lower unknown	Matching Hölder-model lower bound	Unknown	Same KL and first-order model	Hölder first-order oracle $W_2(\mu_0,\pi)=R_0$	No matching result located in the reference corpus.Open literature-check item.	Unverifiedno primary source	—
upper	Averaged LMC	$O(L^2R_0^2/\varepsilon^4)$	$\sqrt{\operatorname{KL}}\le\varepsilon$	Subgradient / LMC step $W_2(\mu_0,\pi)=R_0$	$V$ convex and $L$ -Lipschitz.Checked nonsmooth Euclidean baseline.	Checkedmonograph	Chewi (2026), Theorem 4.3.11 ↗
upper	Nonsmooth mirror-Langevin	$O(L^2D_\phi(\pi,\mu_{0+})/\varepsilon^4)$	$\sqrt{\operatorname{KL}}\le\varepsilon$	Dual-norm subgradient + mirror mapBregman-divergence scale	$\phi$ one-strongly convex in the primal norm.Not directly comparable to the Euclidean $R_0$ rows.	Checkedmonograph	Chewi (2026), Theorem 10.3.28 ↗

Scope

The Hölder exponent interpolates between the smooth case $s=1$ and the Lipschitz/nonsmooth endpoint $s=0$ (up to the usual subgradient interpretation). Mirror rows use their own norm and Bregman geometry and are shown as separate comparators.

Lower bounds

No matching lower bound for the full Hölder-gradient model is recorded in the current reference corpus.